1. Finding Linear Equations Section 1.5

2. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 2 Using Slope and a Point to Find an Equation of a Line Find an equation of a line that has slope m = 3 and contains the point (2, 5). Substitute into the slope-intercept form: Now we must find b Every point on the graph of an equation represents of that equation, we can substitute and Method 1: Using Slope Intercept Example Solution

3. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 3 Using Slope and a Point to Find an Equation of a Line 5 = 3(2) + b Substitute 2 for x and 5 for y. 5 = 6 + b Multiply. 5 – 6 = 6 + b – 6 Subtract 6 from both sides. – 1 = b Simplify. We now substitute –1 for b into y = 3x + b: y = 3x – 1 Method 1: Using Slope Intercept Solution Continued

4. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 4 Using Slope and a Point to Find an Equation of a Line We can use the TRACE on a graphing calculator to verify that the graph of contains the point (2, 5). Method 1: Using Slope Intercept Graphing Calculator

5. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 5 Using Two Points to Find an Equation of a Line Find an equation of a line that contains the points (– 2, 6) and (3, –4). Find the slope of the line: We have Line contains the point (3, –4) Substitute 3 for x and –4 for y Method 1: Using Slope Intercept Example Solution

6. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 6 Using Two Points to Find an Equation of a Line –4 = –2(3) + b Substitute 3 for x. –4 for y. –4 = –6 + b Multiply. –4 + 6 = –6 + b + 6 Add 6 to both sides. 2 = b Simplify. Substitute 2 for b into y = –2x + b: y = –2x + 2 Method 1: Using Slope Intercept Example

7. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 7 Using Two Points to Find an Equation of a Line We can use the TRACE on a graphing calculator to verify that the graph of y = –2x + 2 contains the points (–2, 6) and (3, –4). Method 1: Using Slope Intercept Graphing Calculator

8. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 8 Finding a Linear Equation That Contains Two Given Points To find the equation of a line that passes through two given points whose x-coordinates are different, 1.Use the slope formula,, to find the slope of the line. 2. Substitute the m value you found in step 1 into the equation Method 1: Using Slope Intercept Guidelines

9. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 9 Finding a Linear Equation That Contains Two Given Points 3.Substitute the coordinates of one of the given points into the equation you found in step 2, and solve for b. 4.Substitute the m value your found in step 1 and the b value you found in step 3 into the equation. 5.Use a graphing calculator to check that the graph of your equation contains the two points. Method 1: Using Slope Intercept Guidelines Continued

10. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 10 Using Two Points to Find an Equation of a Line Find an equation of a line that contains the points (– 3, –5) and (2, –1). First we find the slope of the line: We have y = x + b. The line contains the point (2, –1) Substitute 2 for x and –1 for y: Method 1: Using Slope Intercept Example Solution

11. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 11 Using Two Points to Find an Equation of a Line –1 = (2) + b Substitute 2 for x. –1 for y. –1 = + b (2) = ∙ = 5∙(–1) = 5∙ + 5∙b Multiply both sides by 5. –5 = 8 + 5b 5∙ = ∙ = = 1 –13 = 5b Subtract 8 from both sides. – = b Divide both sides by 5. 4545 4545 8585 4545 4545 2121 8585 8585 5151 8585 8181 13.5 Method 1: Using Slope Intercept Solution Continued

12. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 12 Using Two Points to Find an Equation of a Line We can use the TRACE on a graphing calculator to verify that the graph of contains the points (–3, –5) and (2, –1). So, the equation is y = x – 4545 13.5 Method 1: Using Slope Intercept Solution Continued Graphing Calculator

13. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 13 Finding an approximate Equation of a Line Find an approximate equation of a line that contains the points (–6.81, 7.17) and (–2.47, 4.65). Round the slope and the constant term to two decimal places. First we find the slope of the line: Method 1: Using Slope Intercept Example Solution

14. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 14 Finding an approximate Equation of a Line We have y = –0.58x + b. Since the line contains the point (–6.81, 7.17), we substitute –6.81 for x and 7.17 for y: 7.17 = –0.58(–6.81) + b 7.17 = 3.9498 + b 7.17 – 3.8498 = 3.9498 + b – 3.9498 3.22 b Sub -6.81 for x, 7.17 for y. Multiply. Subtract 3.9498 from both sides. Combine like terms. Method 1: Using Slope Intercept Solution Continued

15. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 15 Finding an approximate Equation of a Line We can use the TRACE on a graphing calculator to verify that the graph of comes very close to the points (–6.81, 7.17) and (–2.47, 4.65). So, the equation is y = –0.58x + 3.9498 Method 1: Using Slope Intercept Solution Continued Graphing Calculator

16. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 16 Defining Point-Slope Form Second method to find a linear equation of a line. Suppose that a nonvertical line has: Slope is m y-intercept is (x 1, y 1 ) (x, y) represents a different point on the line So, the slope is: Method 2: Using Point-Slope

17. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 17 Defining Point-Slope Form Given the slope, multiple both sides by x – x 1 gives ∙ (x – x 1 ) = m (x – x 1 ) y – y 1 = m (x – x 1 ) We say that this linear equation is in point-slope form. If a nonvertical line has slope m and contains the point (x 1, y 1 ), then an equation of the line is y – y 1 = m (x – x 1 ) Method 2: Using Point-Slope Definition

18. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 18 Using Point-Slope Form to Find an Equation of a Line A line has slope m = 2 and contains the point (3, –8). Find the equation of the line Substituting x 1 = 3, y 1 = –8 and m = 2 into the equation y – y 1 = m (x – x 1 ). Method 2: Using Point-Slope Example Solution

19. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 19 Using Point-Slope Form to Find an Equation of a Line Use point-slope form to find an equation of the line that contains the points (–5, 2) and (3, –1). Then write in slope-intercept form. First find the slope of the line: Method 2: Using Point-Slope Example Solution

20. Lehmann, Intermediate Algebra, 4ed Section 1.5Slide 20 Using Point-Slope Form to Find an Equation of a Line Method 2: Using Point-Slope Substituting x 1 = 3, y 1 = –1 and m = into the equation y – y 1 = m (x – x 1 ). Solution Continued